Angles and Congruence

Congruence

As I discussed in an earlier post, there are two basic definitions of geometric congruence that are presented to students. The first is based on measurement:

Definition 1. Two objects are congruent if all of their measurements are the same, and in the same order.

That is, two segments are congruent if they’re the same length; two angles are congruent if they’re equally open; two triangles are congruent if their sides and angles have the same measure (in the same order, but in the case of triangles, that’s inevitable); and so on.

The other is based on transformations:

Definition 2. Two objects are congruent if they are related by an isometric transformation.

Sources intended for high school students, particularly textbooks, tend to prefer the first definition; more rigorous and high-level sources tend to prefer the second one.

Angles

Now let’s consider a few definitions of “angle”.

  1. Angles are formed by two rays that begin at the same point or share the same endpoint. The angle measures the amount of turn between the two arms and is usually measured in degrees or radians. (http://math.about.com/od/glossaryofterms/g/AngleDef.htm)
  2. A shape, formed by two lines or rays diverging from a common point (the vertex). (http://www.mathopenref.com/angle.html)
  3. The amount of turn between two straight lines that have a common end point (the vertex). (http://www.mathsisfun.com/definitions/angle.html)
  4. The space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet. (Google’s search dictionary)

In part, the definitions reflect one problem with the measurement definition of congruence. In linear space, we’re initially inclined to think of measurement in terms of linear measurements. But angle measurement is not a linear measurement, it’s a rotational one.

Most graphing at the high school level takes place on the Cartesian coordinate plane. Indeed, as far as I can tell, an increasing number of high school students graduate having seen nothing else. But in mathematics, there are two dominant ways to graph the relationship between two points in a plane: In terms of location on the Cartesian plane, and in terms of absolute distance and rotation on a polar coordinate plane.

In the real world, I think people are more inclined to think in rotational terms as well, at least for short distances. When travelling at ground level through a grid-based city, sure, we tend to think in terms of “go four miles north and then three miles east” (3, 4), but in terms of actual distance, we’re equally or even more likely to think in terms of “it’s five miles northeast of here” (5, 0.927).

Together, segments and angles represent these two basic forms of planar measurement: Segments represent the linear distance between points, and angles represent the amount of rotation.

However, this latter concept is somewhat abstract. What does it mean for two angles to be congruent? Even deeper: What is an angle?

Angles are formed when two lines, rays, or segments intersect. As some of the definitions show, it’s tempting to then define angles in terms of intersecting lines, rays, or segments. That’s reasonable enough, but it’s not mathematically rigorous, and it causes confusion when applying the concept of congruence to angles.

Let’s say we have two segments: AB measures 7 units and BC measures 5 units, and m∠ABC = 45°. Let’s say we have two rays DE and EF, and m∠DEF = 45°. We say that the angles are congruent because they’re the same size, even though segments are not congruent to rays.

Students struggle with separating the concept of angle congruence from the size of the segments used to draw the angle. Use 1″ segments to draw a 45° angle and 3″ segments to draw a 30° angle; even put arrows on the ends of the segments to indicate that they’re rays. Many students will still think that the 30° angle is bigger, because the image itself is bigger.

So I think that defining angles in terms of the sides is confusing, even if it’s convenient early on. I think a significant part of the problem is that an angle isn’t really a concrete thing. Granted, numbers aren’t truly concrete either, but they’re easier to visualize. An angle is a description of a rotation (“the amount of turn”), which is even more abstract.

Because of the ordering of a standard geometry curriculum, and because of this level of abstractness, I think textbooks tend to act as if an angle is a thing instead of a shadow of an action. It is easier to explain “an angle consists of two rays with a common endpoint” than it is to explain “an angle represents the amount of turn between two objects”, and such an explanation doesn’t rely on transformations.

At the same time, though, I think that it’s mathematically reckless to say that “the parts” of an angle are its vertex (the common endpoint) and the sides (the rays). It may be easier at the outset, but it communicates the wrong concepts to the students (in particular, it reinforces the notion that the visual length of the rays or segments is connected to the angle’s size).

Geometry classes also tend to spend a good amount of time on angles without explaining the purpose. Angles are most important because they define polygon similarity, and help define polygon congruence. Yes, there’s also all that stuff about angle congruence between parallel lines, complementary and supplementary angles, and so on, but it feels like we drumbeat the abstract before applying it to the visually (albeit not necessarily “real world”) concrete.

“Isometric transformations” is scary high-level math talk, but the underlying concept is very visually accessible: Draw a shape on a piece of patty paper; move it around, flip it over. No matter what you do with it, that shape and any shadows it leaves behind are congruent.

I think if we started there, or at least gave a preview of that being where we’re going with all the angle talk, it would be easier to be rigorous from the outset: Two angles are congruent if they represent the same degree of rotation.

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